Online Mixed Packing and Covering

In many problems, the inputs arrive over time, and must be dealt with irrevocably when they arrive. Such problems are online problems. A common method of solving online problems is to first solve the corresponding linear program, and then round the fractional solution online to obtain an integral solution. We give algorithms for solving linear programs with mixed packing and covering constraints online. We first consider mixed packing and covering linear programs, where packing constraints are given offline and covering constraints are received online. The objective is to minimize the maximum multiplicative factor by which any packing constraint is violated, while satisfying the covering constraints. No prior sublinear competitive algorithms are known for this problem. We give the first such --- a polylogarithmic-competitive algorithm for solving mixed packing and covering linear programs online. We also show a nearly tight lower bound. Our techniques for the upper bound use an exponential penalty function in conjunction with multiplicative updates. While exponential penalty functions are used previously to solve linear programs offline approximately, offline algorithms know the constraints beforehand and can optimize greedily. In contrast, when constraints arrive online, updates need to be more complex. We apply our techniques to solve two online fixed-charge problems with congestion. These problems are motivated by applications in machine scheduling and facility location. The linear program for these problems is more complicated than mixed packing and covering, and presents unique challenges. We show that our techniques combined with a randomized rounding procedure give polylogarithmic-competitive integral solutions. These problems generalize online set-cover, for which there is a polylogarithmic lower bound. Hence, our results are close to tight

A joint replenishment competitive location problem

Competitive Location Models seek the positions which maximize the market captured by an entrant firm from previously positioned competitors. Nevertheless, strategic location decisions may have a significant impact on inventory and shipment costs in the future affecting the firm’s competitive advantages. In this work we describe a model for the joint replenishment competitive location problem which considers both market capture and replenishment costs in order to choose the firm’s locations. We also present an metaherusitic method to solve it based on the Viswanathan’s (1996) algorithm to solve the Replenishment Problem and an Iterative Local Search Procedure to solve the Location Problem.N/

A Hypergraph Multi-Exchange Heuristic for the Single-Source Capacitated Facility Location Problem

In this paper, we introduce a large-scale neighborhood search procedure for solving the single-source capacitated facility location problem (SSCFLP). The neighborhood structures are induced by innovative split multi-customer multi-exchanges, where clusters of customers assigned to one facility can be moved simultaneously to multiple destination facilities and vice versa. To represent these exchanges, we use two types of improvement hypergraphs. The improvement hypergraphs are built dynamically and the moving customers associated with each hyperedge are selected by solving heuristically a suitably defined mixed-integer program. We develop a hypergraph search framework, including forward and backward procedures, to identify improving solutions efficiently. Our proposed algorithm can obtain improving moves more quickly and even find better solutions than a traditional multi-exchange heuristic (Ahuja et al., 2004). In addition, when compared with the Kernel Search algorithm (Guastaroba and Speranza, 2014), which at present is the most effective for solving SSCFLP, our algorithm is not only competitive but can find better solutions or even the best known solution to some very large scale benchmark instances from the literature

Median problems in networks

The P-median problem is a classical location model “par excellence”. In this paper we, first examine the early origins of the problem, formulated independently by Louis Hakimi and Charles ReVelle, two of the fathers of the burgeoning multidisciplinary field of research known today as Facility Location Theory and Modelling. We then examine some of the traditional heuristic and exact methods developed to solve the problem. In the third section we analyze the impact of the model in the field. We end the paper by proposing new lines of research related to such a classical problem.P-median, location modelling

Consumer choice in competitive location models: Formulations and heuristics

A new direction of research in Competitive Location theory incorporates theories of Consumer Choice Behavior in its models. Following this direction, this paper studies the importance of consumer behavior with respect to distance or transportation costs in the optimality of locations obtained by traditional Competitive Location models. To do this, it considers different ways of defining a key parameter in the basic Maximum Capture model (MAXCAP). This parameter will reflect various ways of taking into account distance based on several Consumer Choice Behavior theories. The optimal locations and the deviation in demand captured when the optimal locations of the other models are used instead of the true ones, are computed for each model. A metaheuristic based on GRASP and Tabu search procedure is presented to solve all the models. Computational experience and an application to 55-node network are also presented.Distance, competitive location models, consumer choice behavior, GRASP, tabu

Location models in the public sector

The past four decades have witnessed an explosive growth in the field of networkbased facility location modeling. This is not at all surprising since location policy is one of the most profitable areas of applied systems analysis in regional science and ample theoretical and applied challenges are offered. Location-allocation models seek the location of facilities and/or services (e.g., schools, hospitals, and warehouses) so as to optimize one or several objectives generally related to the efficiency of the system or to the allocation of resources. This paper concerns the location of facilities or services in discrete space or networks, that are related to the public sector, such as emergency services (ambulances, fire stations, and police units), school systems and postal facilities. The paper is structured as follows: first, we will focus on public facility location models that use some type of coverage criterion, with special emphasis in emergency services. The second section will examine models based on the P-Median problem and some of the issues faced by planners when implementing this formulation in real world locational decisions. Finally, the last section will examine new trends in public sector facility location modeling.Location analysis, public facilities, covering models